bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Modeling the impact of ex-nido transmitted parasites on colony dynamics

Lauren E. Quevillon1,2, , David P. Hughes1,2,3∗, and Jessica M. Conway1,4∗

1Center for Infectious Disease Dynamics, Pennsylvania State University, University Park, PA, USA 2Department of Biology, Pennsylvania State University, University Park, PA, USA 3Department of , Pennsylvania State University, University Park, PA, USA 4Department of Mathematics, Pennsylvania State University, University Park, PA, USA ∗These authors contributed equally.

Infectious disease outbreaks are a common constraint of group that these groups often live in extremely dense living con- living organisms. (: Formicidae) live in ditions with large colony sizes (up to millions of individu- large, dense colonies and are to a diverse range of parasites als; (7), reviewed in (8)), and have an average higher genetic and pathogens, facilitating the possibility of epidemic-induced relatedness compared to other groups. This risk is collapse. However, the majority of parasites infecting ants re- further compounded by nesting habitats in soil and decaying quire a period of development outside of the nest before they wood that put colonies at increased risk of contact with mi- can transmit to their next ant host (‘ex-nido’ transmission) and crobial loads capable of causing disease (9). Additionally, the impact of these parasites on colony dynamics is unknown. Here we develop a mathematical model to assess dy- neighboring colonies overlap in their foraging and namics in the presence of such parasites. We find that under compete for the same food resources, enhancing the potential field-realistic model conditions, such parasites are unlikely to for inter-colony transmission (10, 11). Indeed, social cause the epidemic collapse of mature ant colonies, unless colony are host to a diverse range of pathogens, parasites, and par- birth rate drops below 0.2328 ants/day. The preponderance of asitoids (hereafter ‘parasites’) (12, 13). However, ants and ex-nido transmitting parasites infecting ants and their limited other social insects have achieved incredible ecological suc- epidemiological impact on colony dynamics may partly explain cess (14, 15), and effectively managing their disease burden why collapsed ant colonies are rarely, if ever, observed in natu- could be one underlying reason for their continued success ral populations. over evolutionary time. social insects | ants | epidemiology | parasites Attempting to understand how social insects have contended Correspondence: [email protected] with considerable parasite pressure over their long evolution- ary history has driven empirical research during the past two Introduction decades. This work has uncovered many disease-fighting Infectious diseases capable of massive mortality events ap- mechanisms that social insects have in their arsenal, includ- pear to be an unavoidable consequence of living in large, ing physiological, behavioral, and organizational defenses at complex societies. As humans evolved from small groups of both the individual- and colony-levels (reviewed in (12, 16– hunter-gatherers into larger agrarian societies and then cities, 19)). Some mechanisms, such as allogrooming (the groom- we have seen an increase in the number of outbreaks that our ing of nest mates) (20, 21) and the transfer of immune mod- societies experience (1, 2). Some diseases (i.e. measles, sea- ulators or antibiotic secretions between nest mates (21, 22), sonal influenza) are ‘crowd diseases’ that can only be main- are evolutionary innovations following their transition to eu- tained because of large numbers of individuals living in close , but it is likely that many of these immune mecha- proximity. Outbreaks readily occur in large natural and man- nisms were present prior to the transition to (23). aged animal populations as well. Boom-bust cycles in Gypsy Complementary theoretical studies have used a variety of moths driven by virus epidemics (3), Ebola and anthrax out- modeling frameworks to test the relative efficacies of these breaks amongst chimpanzees and gorillas (4, 5), and recur- mechanisms against parasites that are capable of direct trans- rent foot-and-mouth disease epizootics in commercial bovid mission from one infected nest mate to another. For example, all demonstrate that high density, group living organisms of- deterministic compartmental modeling has been used to as- ten contend with the steep costs of intense disease burden. sess differences in epidemiological outcomes when passive However, infectious disease outbreaks may not necessarily versus active immune modulators are passed between nest be an unavoidable consequence of evolving to live in dense mates (24), and how grooming, a behavioral defense used to groups. Social insects in the order Hymenoptera (ants, , remove infectious particles prior to infection, could impact and ), and the ants in particular, exemplify the liv- epidemic potential inside colonies (25). Others have used ing conditions that exaggerate the perceived risk of infec- individual-based modeling approaches to investigate how in- tious disease spread. These animal groups are defined by teraction heterogeneity (26), nest architecture (27), and im- eusociality, the highest of social organization, in which munity and hygienic behavior (28) impact disease sever- there is a reproductive division of labor, overlapping gener- ity and colony survival. These modeling approaches have ations, and cooperative brood care (6). Eusociality means been very useful for comparing the relative efficacies of anti-

Quevillon et al. | bioRχiv | November 14, 2018 | 1–25 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. disease defenses against parasites that are capable of trans- colony dynamics and show that for a range of biological plau- mission inside the nest. sible parameter values, colony collapse is unlikely to occur. One key assumption underlying most of the above work is Finally we explore how other parameters (i.e. proportion of that parasites infecting ants are capable of direct, ant-to-ant the colony foraging, parasite developmental rate) impact the transmission inside the nest. However, the majority of para- potential impact of ex-nido parasitism on ant colony dynam- site that infect ants have lifecycles that require a pe- ics. Our work suggests that disease dynamics in ant and other riod of development outside of the colony (‘ex-nido’), often social societies, in which ex-nido parasitism predomi- as free-living stages or in other host species, before they are nates rather than parasites directly transmitting between nest able to infect new hosts (Fig. 1) (12, 13). The amount of time mates, are fundamentally different from other social living such parasites need before they are capable of infecting a new organisms like , which may partly explain the en- ant host varies considerably, from days to months or even during ecological success of the social insects. years. For example, following emergence from their ant host, phorid flies must mate and then find new hosts to oviposit Methods into within days (29). Trematodes infecting ants must com- plete development inside their final host, mate and A. Model of ant colony dynamics in the presence and release to be consumed by a snail before another ant can absence of ex-nido transmitting parasites. Our model be infected, a process which can take months (30). Finally, of mature ant colony dynamics in the presence and absence in temperate systems, the zombie-ant fungus Ophiocordyceps of ex-nido parasites is given by Eqs. (1-4) and summarized does not develop the sexual stages needed to release spores schematically in Fig. 2. In the absence of parasitism, we and infect new ants until many months following the death of make the simplifying assumption that there are three devel- its original ant host (31). Thus, with these ‘ex-nido’ parasites, opmental and sociological castes (compartments) that indi- direct transmission between nest mates inside colonies is not viduals transition through in ant colonies, listed below: possible, potentially precluding the threat of major disease outbreaks inside colonies. • Brood (B): a period of biological development during Fig. 1. Transmission strategies used by parasites infecting ants. which individuals transition from through several larval stages.

Direct transmission Indirect transmission • Nest workers (N): a variable period of time during (One host) (Two or more hosts) which individuals participate in intranidal tasks such as brood care and nest maintenance.

ex-nido • Foragers (Fs): a variable period of time that older ants in-nido transition into according to colony need; foraging ants gather food and participate in extranidal de-

ex-nido fense and maintenance, which puts them at greater risk for both and parasitism.

days weeks We believe that this captures sufficient biological realism of months mature ant colony functioning without over-complicating the require development outside of the nest before transmission to new hosts model. While some ant species do have other specialized (a) Direct, in-nido parasitism occurs when ants are the only host infected and trans- castes (e.g. soldiers in army ants, repletes in honey pot ants, mission can readily occur from ant-to-ant inside the nest. Examples include bac- etc.), the basic colony demographic structure we have se- teria, viruses, and some entomopathogenic fungi. (b) Direct, ex-nido parasitism occurs when ants are the only host infected, but the parasite needs to complete lected applies broadly to the majority of ant colonies (10, 32). development in the extranidal environment before it is capable of transmitting to Our investigation of colony dynamics in the presence of ex- the next ant host. Examples include parasitoid wasps and flies. (c) Indirect, ex-nido nido parasites, i.e. parasites that require a developmental pe- transmission occurs when more than one host species is required for the parasite to complete its lifecycle. The use of multiple hosts necessitates development outside riod outside of the nest before transmission to new hosts, is of the nest. Examples include cestode, trematode, and some worms. motivated by the biology of the most prevalent of ant- infecting parasite that we find from our extensive review of the literature (13), parasite records in (12, 29, 33–37), among To date, no theoretical studies have investigated the potential many others). We model a parasite that encounters and in- impact of ex-nido parasites on ant colony dynamics. Here, fects a forager ant in the extranidal (outside of the nest) en- we formally explore the consequences of ex-nido parasites vironment, causes the mortality of that ant after some devel- by building a model of ant colony dynamics in the absence of opmental period, and ultimately transmits to its next host in parasite pressure; we then extend it to a ‘susceptible-infected- the extranidal environment (i.e. no direct ant-to-ant transmis- removed’ (SIR)-type model for the disease dynamics of par- sion within the nest). Thus, our model of colony dynamics asites transmitted outside of ant colonies. While this model in the presence of parasitism includes a fourth compartment, was built to capture ant colony dynamics, it is generally ap- infected foragers (Fi). When the force of infection term β plicable to other social insect colonies. We explore how (described in detail below) is zero, Eqs. (1 - 3) reduce to the changing colony birth and parasite transmission rates impact uninfected model, which is used to validate parameter choice

2 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. A Model of ant colony dynamics in the presence and absence of ex-nido transmitting parasites and the form of the model. When the force of infection term (2-3)). We explore how sensitive colony population dynamics β is greater than zero, Eqs. (1 - 4) represent the infected are to changing this forager proportion κ under the presence model. of ex-nido parasitism in Fig. 5. We also perform sensitivity Fig. 2. Schema for the model of colony growth in the absence and presence of analyses for predicted colony dynamics under changing α in ex-nido parasitism. Fig. S4.

dN F + F Model schemata = φB − αN(κ − s i ) − µN (2) dt 1 + Fs + Fi + N + B

β γ dFs Fs + Fi Fs Fi = αN(κ − ) − µFs − βFs (3) dt 1 + Fs + Fi + N + B μ μ Once nest workers become foragers, we make the simplify- α, σ ing assumption that they remain foragers until death, which serves to reduce model complexity. For many ant species, we N do not know exactly how workers are allocated to the task of μ foraging and how much task fidelity these workers exhibit. It is generally assumed that social insect colonies have some φ degree of temporal polyethism, in which workers age through a sequence of behavioral castes (10, 38) with foraging usually B occurring later in life. How workers cycle through tasks over μ their lifetime likely depends on their physical caste as well (38, 39). While temporal polyethism gives a general pattern λ to how individuals cycle through tasks during their lifetime, uninfected model behavioral flexibility allows colonies to be resilient during catastrophes (40–42), further complicating how colonies are Model compartments and flow. Brood (B) are born into the colony according to birth socially organized. For modeling simplicity, we make the as- rate λ. Brood transition to become nest workers (N) according to the developmental rate, φ. Nest workers transition to become susceptible foragers (Fs) according to sumption that workers follow the traditional series of tasks colony need (κ) by rate α. Finally, susceptible foragers can become infected (Fi) over their lifetime (interior tasks as nest workers, followed according to the parasite force of infection term, β. All compartments experience by exterior tasks as foragers) and that foragers remain in that the same natural mortality rate, µ. compartment for life. We leave investigation of the impact of behavioral flexibility observed in some colonies on interac- New ants are born into the brood compartment B with colony tion with ex-nido parasites for future work. birth rate λ via a Hill function (see Eq. 1), which we assume dF i = βF − µF − γF (4) is dependent on the total colony size and minimum colony dt s i i size σ needed to maintain colony viability. We choose a In the presence of parasitism (β > 0), susceptible foragers Hill function to make the number of brood born into the nest F become infected by parasites in the environment at rate conditional on the total colony size, because workers sup- s β, to become infected foragers F (see Eq. 4). The per- port brood care by, for example, gathering food resources, i capita rate β is a parameter that combines the frequency with feeding larvae, and cleaning brood (10). Thus, colonies with which foragers leave the nest, their probability of encoun- more workers can support greater numbers of brood. Brood, tering parasites whilst outside the nest, and the probability B, can become nest workers, N, after a developmental period of successfully becoming infected given an encounter with (transitioning at per-capita rate φ) or can die due to natural a parasite. Like their uninfected counterparts, infected for- mortality (at per-capita rate µ). We explore a range of colony agers F can die due to natural mortality µ; however, they birthrates to assess the impact of the λ parameter on colony i will more likely die a parasite-induced death at rate γ, which dynamics under the presence of ex-nido parasites (Fig. 4). occurs at a more rapid rate (see Table 1 for model param- 2 eters). Though these infected foragers die at a more rapid dB λ(Fs + Fi + N + B) = 2 − φB − µB (1) rate than their uninfected counterparts, they do not pass on dt σ2(1 + (Fs+Fi+N+B) ) σ2 the infection, and in other respects interact with the colony Nest workers can either die due to natural mortality at per- like uninfected foragers. Note that infected foragers are in- capita rate µ or transition into the forager compartment, Fs, cluded in the calculation of the total colony size and we as- according to colony need. Here we make the assumption that sume that there are no infection-related changes in individual a fixed proportion, κ of the colony is needed for extranidal or colony-level behavior. In addition, infected foragers are tasks. We model the nest worker to forager transition using a always assumed to die (there is no recovery from infection), logistic growth-like term, multiplied by the transition rate α, which is realistic given that most parasites known to infect which gives the maximum per-capita transition rate (see Eqs. ants require ant death as developmental necessity (13). For

Quevillon et al. | Modeling ant parasites bioRχiv | 3 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Model parameters Parameter Unit Definition Modeling range Baseline value Lambda, λ ants/day Colony birth rate 1 - 100 10 Phi, φ days−1 Brood - nest worker transition rate - 1/56 Alpha, α days−1 Nest worker - forager transition rate 0.1 - 1 0.1 Kappa, κ - Max proportion of the colony in the forager compartment 0.1 - 1.0 0.3 Sigma, σ ants Minimum colony size - 10 Mu, µ days−1 Natural mortality rate 1/760 - 1/30 1/365 Beta, β days−1 Per-capita parasite transmission rate 0.0001 - 1 0.01 Gamma, γ days−1 Parasite-induced mortality rate 1/30 - 1 1/7

Table 1. Description of model parameters, the range of their values, and the baseline values used in the modeling analysis.

most results we assume that the parasite is constantly present and infected foragers (Fs +N +B,Fs +Fi +N +B for β = 0 in the environment, though we also explore the potential im- and β > 0 cases, respectively), but do not take seasonal fluc- pacts of seasonality in the force of infection term β, which tuations into consideration. Many mature ant colonies fluctu- may arise due to seasonal fluctuations in parasite population ate in their colony sizes and birth rate seasonally (10). Most size or prevalence. We also formally explore the impacts of of this seasonal fluctuation is due to the seasonal production changing the parasite-induced mortality rate in Fig. S3. of reproductive forms; these fluctuations are species-specific and correspond to the synchronous production and release B. Additional model assumptions. of reproductives for mating flights between colonies. Since we do not include reproductive forms in our model, we do Compartments experience identical natural mortality rates.. not include seasonality in our birth rate parameter λ, which We assume that ants in all compartments experience an iden- also has the benefit of rendering our model more analytically tical rate of natural mortality µ. Empirical estimates of tractable. worker mortality rates are scarce in the literature (Table S4). While we would expect younger ants (brood and nest work- Only foragers are exposed to ex-nido parasites.. We are in- ers) to have lower natural mortality than foragers, due to their vestigating parasites infecting ant colonies via an ex-nido comparative youth and performance of less-risky tasks within mode of transmission, which requires extranidal parasite ex- the nest, that expectation has not yet been confirmed empir- posure in order to become infected (13). In many ant soci- ically, nor have the relative mortality rates been quantified. eties, this means that only foragers are at risk because only Therefore, we have made the simplifying assumption that all foragers leave the relatively protected confines of the nest. ants have the same lifespan of one year, corresponding to a However, some exceptions to this do occur. For nomadic natural mortality rate of 1/365 days−1. We explore the effects species such as army ants, periodic relocation makes all ants, of changing the natural mortality rate in Fig. S2. including brood and reproductives, susceptible to infection, particularly from parasitoids, which can conduct aerial at- No reproductive castes are included.. We do not include any tacks on foraging ants (43). In other cases, parasites them- reproductive castes (unmated males (drones), unmated fe- selves are able to enter the nest independently of the behavior males (), or mated females (queens)) as explicit com- of ant workers, either through mobile stages actively entering partments in our model of mature ant colony dynamics. Re- the nest (e.g. Syrphid fly microdon larvae) or through par- productives do not follow the same demographic flow (i.e. asitoids directly ovipositing their young inside the nest (e.g. brood to nest worker to forager) as non-reproductive work- some Phorid fly species). For a first attempt at understanding ers and do not participate in either intranidal or extranidal how ex-nido parasites impact ant colony dynamics, we have working tasks (10). Furthermore, gynes and drones are ex- chosen to only focus on the case where foragers are exposed pected losses for the colony; they leave on their mating flight to parasites. In future work, we hope to address the other and do not return, so whether they become infected once they ways that ex-nido transmitting parasites impact ant colony have left the nest is inconsequential for colony survival (al- growth. though this does have important consequences for colony fit- ness). The queen is implicitly included in our model, as she No seasonality in the parasite force of infection, β.. For mod- is responsible for the birth of new ants into the brood com- eling simplicity, we assume that the parasite force of in- partment. However, as she is a singular individual (or a few fection term β is constant, though empirical evidence sug- individuals, in the case of polygynous colonies), we do not gests that parasites infecting ants are heterogeneous in both create a separate compartment for her. time and space (44, 45). To assess whether seasonality in the parasite force of infection term might improve colony No seasonality in colony birth rate.. The colony birth rate λ is rescue role, we further investigated how one or two annual assumed to depend on the total colony population for which peaks in β could impact infection dynamics. To do this, we we take into account all model compartments including brood use standard approaches (e.g. like that used in (46)) and

4 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. C Baseline model validation assume that the force of infection term β takes the form compartment within the infected model (β = 0.01 days−1) β(t)= β0(1 + β1 sin(2πt)) or β(t)= β0(1 + β1 sin(4πt)), under baseline parameter values (Table 1) are Fs: 458.134 where t is in years but is adjusted to match our time scale, ants, Fi: 31.4659 ants, N: 1,034.09 ants, and B: 485.499 ants. to get one or two annual peaks, respectively. We explore the Bifurcation diagrams for the model when β = 0 (uninfected) impacts of seasonality in Figs. 6, S5, and S6. and when β > 0 (infected) are provided in Figs. 3a and 3b, respectively. In Fig. 3a, the bifurcation diagram switches Model parameters. A description of the model parameters from a stable to unstable regime at the minimum colony size and their values are given in Table 1. Whenever possible, es- needed for functioning (min = 10 ants). In Fig. 3b, the bifur- timates of model parameters were extracted from published cation diagram with respect to β remains stable throughout values. We provide a detailed discussion of how model pa- the range of β values for a colony birth rate λ = 10 ants/day. rameter values were chosen and summarize empirical esti- Fig. 3. Bifurcation diagrams with respect to colony birth rate λ and parasite force of mates from the literature in the Supplemental Information infection, β. and the tables therein. Bifurcation diagram: λ Results 20,000

We are ultimately interested in the impact of ex-nido para- 17,500 stable sitism on ant colony dynamics under biologically realistic unstable 15,000 parameter space. To that end, here we first provide valida- tion of our model in the absence of parasitism, and then show 12,500 how our model predicts parasite-induced impacts on colony (ants) 10,000 dynamics under changing values of colony birth rate and par- 7,500 Total colony size Total asite force of infection. Finally, we assess model sensitivity 5,000 to different parameter values such as changing the proportion 2,500 of the colony foraging, changing the natural mortality rate, and adding seasonality to the parasite force of infection. 0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 log λ (ants/day) C. Baseline model validation. To understand the effect of 10 ex-nido parasitism on ant colony dynamics we first exam- Bifurcation diagram: β ined dynamics in the absence of parasitism (force of infec- −1 tion β = 0 days ), using baseline parameter values given in 3,500 stable Table 1. Using linear stability analysis and confirmation via unstable 3,000 direct numerical simulation, we found that for all birth rates in the range of modeled values, colony growth reaches equi- 2,500 librium in approximately 1,500 days, which corresponds well 2,000 (ants) to knowledge of ant colony development from inception to 1,500

colony maturity (8, 10, 32, 47, 48). To assess how the model colony size Total 1,000 reacts to population perturbations in the absence of ex-nido parasitism, we simulated the removal of a fixed percentage 500 0 of each compartment (B, N, Fs) after equilibrium had been -1.5 0.0 reached (Fig. S1). In all cases, as long as the population had -4.0 -3.5 -3.0 -2.5 -2.0 -1.0 -0.5 log β (day-1) not been reduced below the minimum value necessary to sus- 10 tain colony functioning (σ, Table 1), each compartment was (a) Bifurcation diagram for the uninfected model (eqs. 1-3, β = 0) under baseline parameter values given in Table 1. Rescaled colony size as a function of colony birth able to recover back to their equilibrium values. The time it rate. Colony size was rescaled using an arcsinh function. (b) Bifurcation diagram took to recover from the perturbation to 90% of equilibrium for the infected model (eqs. 1-4) under baseline parameter values given in Table 1. values was 600 - 820 days (2 - 3 years) and 1,110 - 1,650 days to return to full equilibrium values. While it is unknown how long it would take an ant colony to recover back to pre- E. Changing birth rate, λ, and force of infection, β. We perturbation population values in a natural setting, it seems next examined colony dynamics in the presence of ex-nido realistic that the recovery period is on par with the growth parasitism. Under the baseline colony birth rate and force of period of an incipient ant colony reaching maturity. infection given in Table 1, we observe an equilibrium reduc- tion of 45.8% in colony size in the infected model relative to D. Equilibrium values and bifurcation diagrams. The the uninfected model (i.e. β = 0 days−1). It takes approx- equilibrium values for each compartment within the unin- imately 2,900 days to reach this full equilibrium reduction fected model (β = 0 days−1) under baseline parameter val- and approximately 800 days to be within 10% of equilibrium ues (Table 1) are as follows- Fs: 1,045.91 ants, N: 2,118.56 values. When β is fixed at the baseline value of β = 0.01 ants, and B: 485.507 ants, which is a realistic colony size for days−1 and we assume different values of colony birth rate many ant species (7, 10). The equilibrium values for each λ from 1 - 100 brood/day, we predict that the proportion re-

Quevillon et al. | Modeling ant parasites bioRχiv | 5 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. duction in colony size remains essentially the same (Fig. 4a), the uninfected model (Fig. 5a) and on the total colony size and it takes 700 - 900 days to be within 10% of equilibrium at equilibrium (Fig. 5b). When 10% of the colony forages values. When we fix colony birth rate λ at the baseline value under baseline model conditions, there is a 22.8% reduction of λ = 10 brood/day, chosen because it’s in the middle of log in colony size compared to the uninfected model; in contrast, scale realistic values for colony birth rates (see Table S1) and when 70% of the colony forages, there is a 61.3% reduction traverse values of β in the range of β = 0.0001 to 1 day−1, in total colony size relative to the uninfected model (Fig. 5a). we find a wide range in resulting proportion reductions in This translates into a large difference in resulting total colony colony size, from 1% up to 77%, respectively (Fig. 4a). This sizes for a typical ant colony: 2,816 ants vs. 1,412 ants, for is realistic because β controls the rate at which foragers are κ = 0.10 vs. 0.70, respectively. By allowing a larger propor- lost due to parasitism; above a certain value of β the colony’s tion of the colony to be in the susceptible forager (Fs) com- birth rate can no longer compensate for the loss of foragers partment, more individuals are at risk for becoming infected, and thus larger reductions in colony size occur (Fig. 4). and so a larger number of individuals die due to parasite- Fig. 4. Predicted impact of ex-nido parasitism on colony growth dynamics. induced mortality rather than natural mortality. Our model, which makes the simplifying assumption that foragers expe- A rience the same natural mortality as their non-foraging coun- % colony size reduction relative to β = 0 terparts, likely overestimates the percent relative reduction in 2.0 100 colony size between infected and uninfected models because 90 foragers likely do experience a higher rate of natural mortal- 1.5 80 ity. Fig. 5. Sensitivity of model predictions to changing the proportion of the colony 1.0 70 foraging, κ. λ 60 10 0.5 % (ants/day)

log 50 Sensitivity of model predictions to changing proportion of the colony foraging, κ 0.0 40

30 −0.5 20

10 −1.0 Colony collapse β = 0.01 (baseline) 0 baseline κ value −5 −4 −3 −2 −1 0 1 log10β (days-1) Traversing β Traversing λ relative to uninfected model β = 0.001

(force of infection) (colony birth rate) in colony Percent reduction size 100 0.1

80 1.0 a 1,10,100 0.1 60 Proportion of the colony foraging, κ 0.01 40

20 0.001 % Relative Reduction % Relative Reduction 0.0001 0 β = 0.001 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 log β 10 log10λ (days-1) (ants/day) β = 0.0001 λ = 0.1 β = 0.001 λ = 1 (ants/day) β = 0.01 (days-1) λ = 10 β = 0.1 β = 0.01 (baseline) λ = 100

β = 1 colony size Total (a) ContoursB in percent reduction in colony size predicted by our model (Eqs. (1-C baseline κ value 4)) as a function of colony birth rate λ and force of infection β relative to when β = 0. Other parameters can be found under baseline values in Table 1.(b) Percent reduction in colony size predicted by our model (Eqs. (1-4)) as a function of rate of infection β and λ = 10 brood/day relative to when β = 0. Other parameters can be found under baseline values in Table 1.(c) Percent reduction in colony size predicted b by our model (Eqs. (1-4)) as a function of colony birth rate λ and β = 0.01 relative Proportion of the colony foraging, κ to when β = 0. Other parameters can be found under baseline values in Table 1. Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under changing values of κ from 0 to 1. (b) Total colony size (Fs + N + B) under changing values of κ from 0 to 1. F. Impact of the proportion of the colony foraging, κ. The proportion of the colony that forages κ is an important model parameter because it impacts the number of individ- G. Adding seasonality to parasite transmission rate. uals in the susceptible forager Fs compartment and thus the To assess how seasonality in the parasite force of infection number of individuals at risk of becoming infected. Altering term β could impact colony dynamics, we simulated out- the proportion of the colony foraging κ had a large impact breaks in which β varies over time rather than being held at a on the resulting percent reduction in colony size relative to fixed value. To accomplish this, in these simulations the force

6 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. H Changing the natural mortality rate µ

of infection term β takes the form β(t)= β0(1+β1 sin(2πt)) the parasite-induced mortality rate γ and the natural mortal- or β(t)= β0(1 + β1 sin(4πt)), to get one or two seasonal ity rate µ, resulting in a larger disparity when comparing the peaks per year, respectively. We plot these seasonally varying colony population sizes of the infected and uninfected mod- values of β and their impacts on the percent colony reduction els. In the uninfected model, fewer individuals are dying due relative to the uninfected model in Figs. S5 and S6 and we to natural mortality, so there is a larger population size, while show a comparison of model predictions for total colony size population loss in the infected model is being driven primar- in cases where β is either constant, has 1 seasonal peak per ily by parasite-induced mortality. year, or has two seasonal peaks per year in Fig. 6. When β reaches a seasonal peak, colony sizes in all three cases ap- I. Changing parasite-induced mortality rate, γ. Chang- proximately mirror each other. However, including seasonal- ing γ, the parasite-induced mortality rate for infected for- ity in the force of infection term β allows colony size to re- agers, alters the equilibrium number of infected foragers. bound slightly, resulting in a higher average colony size over In our model, infected foragers are assumed to behave as time Fig. 6. Thus, our results in Fig. 4a, in which parasitism healthy foragers, don’t infect others, and are considered part is assumed to be constant, represent the ‘worst-case’ scenario of the total population size that is used to calculate the frac- for a given value of β. tion of the colony that is foraging. Thus, when the parasite- Fig. 6. Comparing the impact of constant vs. seasonal parasitism. induced mortality γ is lower (but still faster than the rate of death due to natural mortality), infected foragers survive Comparing the impacts of constant vs. seasonal parasitism longer and replacements do not need to be recruited from the on total colony size nest worker population as frequently, which helps reduce the size of the forager population that is exposed to infection at

4,000 no parasitism parasitism present rate β. However, this difference has only a small impact on the percent reduction in colony size relative to the uninfected β = 0 β = 0.01 (baseline) model (Fig. S3a) and the total colony size at equilibrium (Fig. 3,000 β = 0.01, 1 peak −1 max S3b). When γ is 1/30 days , the resulting percent popu- βmax = 0.01, 2 peaks lation reduction is 43.3%, while the percent reduction only −1 2,000 increases slightly to 46.6% when γ is changed to 1 days

Total colony size Total (Fig. S3a). Thus, model outcomes are not very sensitive to changes in γ within biologically reasonable ranges. 1,000

J. Conditions for colony collapse. Given specific colony

0 0 2,500 5,000 7,500 10,000 parameters, we can use our model to predict colony size re- duction or collapse. In the case of the baseline parameters Time (days) (a) Values of the force of infection term β under constant and seasonal parasitism (Table 1), colonies are vulnerable to collapse when the birth with one or two peaks. The maximum values of β = 0.19 days−1. (b) Model rate λ is less than 0.2328 ants/day, which we calculate via predictions of colony size when there is no parasitism present, parasitism is present linear stability analysis of Eqs. (1 - 4) in the limiting case of under baseline values that are constant, parasitism fluctuates seasonally with one annual peak, or parasitism fluctuates seasonally with two annual peaks. Model β → ∞. That is, if λ is above this threshold then collapse will predictions are from numerical simulation under baseline parameter values given in not occur, though the extent of colony size reduction will de- −1 Table 1 and β = 0.19 days . pend on the strength of the force of infection β (Fig. 4a,b). In Fig. 4b, the only value of λ that results in collapse is less than this threshold, and in Fig. 4a, the upper boundary of the col- H. Changing the natural mortality rate µ. As the natural lapse region asymptotically approaches this value. If λ is less mortality rate decreases (i.e. individuals live longer), the to- than the threshold value of 0.2328 ants/day, the colony may tal population increases markedly (Fig. S2b). When ex-nido collapse if β is sufficiently large. Though the total colony size parasites are present (β > 0), changing the natural mortality at equilibrium is sensitive to changing values of κ, the birth rate can also cause large differences in the percent reduction rate threshold above which collapse will not occur is not sen- in total colony size between the infected and uninfected mod- sitive to changing values of κ. For example, for κ = 0.01, els (Fig. S2a). For example, when we take an average ant the minimum value of λ is 0.0656 brood/day, whereas for −1 lifespan of 1 year (µ = 1/365 days ), which has been re- κ = 0.5, the minimum value of λ to avoid collapse is 0.2736 ported for worker ants, the percent relative reduction in total ants/day. While ant colony birth rates are species-specific colony size is approximately 44%, whereas when the aver- and likely do vary seasonally, a birth rate of 0.2328 ants/day −1 age lifespan is assumed to be 30.4 days (µ = 1/30.4 days ), is much lower than those reported in the literature (Table S1). the percent relative reduction is approximately 3%. An av- For much of the parameter space explored in our model (Fig. erage longer lifespan increases the relative impact of para- 4a), colony collapse is not predicted to occur. sitism on the colony in two ways. Firstly, foragers live longer and thus have more cumulative exposure to parasites over the course of their longer lifespan, resulting in a larger number Discussion of infected foragers at equilibrium. Secondly, as the lifespan The ants and other social insects are complex societies that of individuals increases, there is a larger disparity between have evolved over long periods of evolutionary time (6). For

Quevillon et al. | Modeling ant parasites bioRχiv | 7 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. the ants we know this to be between 139 - 158 million years, of β suggests that the parasite infective pressure that ants which parallels major changes in ecological complexity, such face is probably quite heterogeneous in both time and space as the radiation of the angiosperms and expansion of other (52–54). For modeling simplicity we have assumed, for our insect groups (49). During this time period, ants evolved main results, that parasite pressure is constant, but many par- to be the dominant in most terrestrial biomes; de- asites of social insects undergo seasonal fluctuations in their spite accounting for 0.4% of the estimated 5.5 million in- populations (45, 55, 56). When we investigated seasonality sect species (50), they typically comprise > 50% of animal by allowing for one or two annual peaks in parasite force of biomass (15, 51). This success seems paradoxical, as ants are infection β, colonies were able to rebound and colony size host to diverse parasites (12, 13, 33) and seem to be particu- were slightly larger on average (Fig. 6). Our model, which larly vulnerable to infectious disease outbreaks due to living assumes a constant rate of parasite infection, therefore repre- in crowded conditions with lots of highly related individuals. sents an upper bound to the impact of ex-nido parasite pres- Accordingly, a lot of research has been devoted to uncovering sure on ant colony size. Additionally, we know that ants are potential mechanisms by which ants and other social insects able to behaviorally mitigate their potential exposure to para- might mitigate potential disease spread inside their colonies sites by reducing foraging (57–59) and can prevent exposure (reviewed in (12, 16, 18, 19)). However, the majority of the from developing into infection via self- and allogrooming be- parasites that infect ants and social insects use lifecycles that haviors (25, 60). Our model, which does not include be- preclude intra-colony transmission, instead requiring periods havioral flexibility or behavioral anti-parasite defenses, likely of time outside the nest for development in the environment overestimates the impact of parasitism. Thus, the conditions or in other hosts before the next transmission event to ants that lead to model-predicted colony collapse provide a worst- can occur (12, 13). The impact of these parasites on colony case scenario for the colony, and these conditions are unlikely dynamics has been hitherto unexplored. to occur in natural settings. Here we formally explored the potential for epidemic col- Furthermore, the timing over which these parasite-induced lapse due to these ex-nido parasites by modeling the impact colony size reductions occur plays an important role in of their infection on ant colony dynamics. Using parameter colony robustness; these reductions take place over years, values extracted from the literature (Appendix B), we found not in a matter of days (approximately 800 days or over 2 that in the absence of parasitism, colonies can recover from years to be within 10% of full equilibrium reduction under population perturbations back to equilibrium values within baseline parameters), causing a gradual thinning of colony approximately 1,100 - 1,650 days (Fig. ??), which is in gen- size. This contrasts with what we observe in high density, eral accord with reported growth rates of colonies as they ma- group living mammals where a sudden catastrophic loss of ture (47). Such perturbations might reflect attack by preda- individuals can occur, like that recently observed during the tors or competition with other colonies. We then modeled Saiga antelope epizootic (61) or occurring in tropical forest the presence of ex-nido parasites and showed that for a wide populations due to anthrax (62). In natural settings, range of biologically plausible birth rates the effect of para- ant colonies might be able to adapt to or mitigate this grad- sitism ranged from negligible impacts on colony population ual parasite-induced loss of workers before collapse occurs, (< 10% colony size reduction, Fig. 3a) to significant reduc- through changing behavioral defenses (57), moving nest lo- tions (>90% colony size reduction, Fig. 3a), depending on cations (63), or potentially increasing birth rate in compensa- the magnitude of the force of infection parameter β. Un- tion. der the baseline parameters used in our model, colonies are What happens in cases where colony size is reduced by par- not vulnerable to parasite-induced collapse unless λ is be- asitism but collapse does not occur? Few studies have ex- low 0.2328 ants/day, which is much lower than birth rates plored the impact of parasite-induced colony size reduction reported in the literature (Table S1). Thus our modeling pre- on colony functioning, organization, or fitness. Ants and dicts that under realistic parameter values, infection by ex- other social insect colonies do have built-in redundancy to nido parasites will not result in colony collapse. buffer loss (10, 32, 64). Schmid-Hempel and Heeb (65) An important question is whether values of β large enough to found that imposing an extra 10-15% weekly mortality rate cause collapse could be achieved. A key result of this study on Bombus lucorum colonies did not alter colony growth rate is that the impact of ex-nido parasitism depends most heav- or reproductive output compared to control colonies. Follow- ily on the force of infection β, a parameter that combines up work by Müller and Schmid-Hempel (66) experimentally the frequency with which foragers are in the extranidal envi- removed 10% of the worker force each day in B. lucorum ronment, how frequently foragers encounter parasites while colonies and found no difference in worker production or the in that environment, and the probability of successful infec- timing of reproduction but there were reductions in either the tion given contact with a parasite. While few if any empirical number of males produced or female reproductive body size estimates of the components of β are known (see additional depending on when worker loss occurred. One proxy for for- discussion in SI), if foragers make trips very often, encounter ager loss may be circumstances where foragers do not leave parasites very frequently, and/or are unable to prevent suc- the colony for resources because of risk of death from abi- cessful infection by behavioral, social, or physiological im- otic factors like heat. In a long term study on the desert mune mechanisms, then such values could indeed occur. The foraging ant Pogonomyrmex barbatus, Gordon (67) showed little information we do have on the individual components that colonies could reduce the number of foraging trips to

8 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. J Conditions for colony collapse avoid worker desiccation and still achieve colony growth and of their parasites (76), which remains an exciting and under- colony fitness (number of daughter colonies produced). explored area of research. Our work here shows that para- Are there any examples of parasites causing epidemic colony sites that require leaving the nest before the next transmission collapse in natural populations? For the ants, to our knowl- event to ants can occur are likely to have a limited impact edge there are no reports of parasite-induced colony collapse on mature colony dynamics, precluding epidemic collapse. for mature colonies, but there are examples of colonies per- This then may be a key difference between social insect so- sisting despite the multi-year presence of ex-nido transmit- cieties and other animal societies, where massive outbreaks ting parasites (54). While attempts to use both in-nido and ex- of infectious diseases are routine and destabilizing to group nido transmitting parasites as biocontrol agents against pes- functioning. tiferous ant species have been numerous (see review by (68)), ACKNOWLEDGEMENTS their lack of success highlights how difficult it is to perturb We would like to thank Timothy C. Reluga for technical guidance in modeling anal- a mature ant colony to collapse (69–71). Parasite pressure ysis and for insightful discussions thereof. We would also like to thank members of Penn State’s Center for Infectious Disease Dynamics for helpful comments and is most certainly a significant selective pressure on incipient feedback on this work. We gratefully acknowledge the many natural historians who colonies (10), however; this is extremely hard to observe in have collected records of parasites infecting ants as well as the myrmecologists who have studied ant life history traits. Their work provides the biological foundation on field settings and so associated data is extremely limited. which this and other studies can be built. Some social insects are famously experiencing collapse. This material is based upon work supported by the National Science Foundation (NSF) under Grant No. 1414296 to DPH as part of the joint NSF-NIH-USDA Ecol- Managed colonies and global bee populations have been ogy and Evolution of Infectious Diseases program and under NSF Grant No. DMS- experiencing significant decline beginning in the early 2000’s 1714654 to JMC. LEQ is supported by a NSF Graduate Research Fellowship under Grant No. DGE1255832. Any opinions, findings, and conclusions or recommen- (72, 73), but it appears that a constellation of causes includ- dations expressed in this material are those of the authors and do not necessarily ing pesticide usage, habitat loss, chronic stress, and directly reflect the views of the National Science Foundation. transmitting parasites coupled with ectoparasites (mites) con- tributed synergistically to the reported losses (74). 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Quevillon et al. | Modeling ant parasites bioRχiv | 11 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Supplementary Note 1: Model Parameters A description of the model parameters and their values are given in Table 1. Whenever possible, estimates of model parameters were extracted from published values. Here we provide a detailed discussion of how model parameter values were chosen, and summarize empirical estimates from the literature in Tables S1, S2, S3, S4, S5, S6, and S7.

Colony birth rate λ. For colony birth rate, there is a large range in published values in the literature. This is due to large differences between ant species, with some species having extremely large colony sizes and thus large numbers of eggs laid per day while other species have small colonies and smaller birth rates (Table S1). In our model, the overall rate at which new brood are born into the colony is dependent on the colony birth rate parameter λ, the current colony population (Fs + Fi + N + B), the minimum colony size σ needed to maintain colony viability, and this together is modulated via a Hill function (see Eq. 3.1). We explore a range of colony birthrates to assess the impact of the λ parameter on colony dynamics under the presence of ex-nido parasites (Fig. 4).

Brood - nest worker transition rate φ.. For the brood-nest worker transition rate, φ, the range in this parameter is due to the impacts of temperature on brood development time, with values of 18 days to 180 days reported in the literature (Tables S2, S2b). For simplicity, we assume a constant transition rate with value 1/56 days−1, the inverse of the median brood development times reported in the literature.

Nest worker - forager transition α and κ. The nest worker-forager transition is governed by two parameters in our model: κ (the maximum proportion of the colony that are foragers), which dictates how many new foragers need to be recruited from the nest worker compartment, and α, the maximal rate at which the nest worker-forager transition occurs in the absence of foragers. Estimates of the proportion of ant colony populations that engage in foraging are scarce (Table S3), and the estimates that have been reported vary considerably, ranging between 4.5 - 90%. It is likely that the proportion of the colony that foragers changes seasonally, but for model simplicity we make the assumption that a maximum of thirty percent of the colony is in the forager compartment at any given time. We explore how sensitive colony population dynamics are to changing this forager proportion κ under the presence of ex-nido parasitism in Fig. 5. We also perform sensitivity analyses for predicted colony dynamics under changing α in Fig. S4.

Natural mortality rate µ. Our understanding of natural mortality rates and worker lifespan in ants is limited by the relative difficulty of following individuals over the course of their lifetime under field settings. Accordingly, most estimates of worker lifespan come from laboratory studies, which preclude many natural causes of death (i.e. predation, interspecific competition and aggression, and parasitism). Worker life spans greater than one year have been reported for several species (Table S4), with some exceeding three years. For studies that have followed worker life spans in the field, the life spans of ants once they become foragers are far less (6 - 30 days). For simplicity, we assume an individual’s lifespan to be 1 year, which corresponds to a natural mortality rate of 1/365 days−1. We assume that each compartment has the same natural mortality rate; we explore the effects of changing the natural mortality rate in Fig. S2.

Parasite-induced mortality rate γ. In our model, we assume that successful infection leads to mortality, thus we define parasite-induced mortality rate as the rate at which infected individuals die, not their probability of dying. Few estimates of parasite-induced mortality rates for parasites of social insects exist in the literature. We summarize reported estimates of the inverse survival time in Table S5 and formally explore the impacts of changing the parasite-induced mortality rate in Fig. S3.

Supplementary Note 2: Additional modeling results Changing the nest worker - forager transition term α. Changing the nest-forager transition term α does not qualitatively change either the transient or equilibrium dynamics for the model (Fig. S4). An α of 0.1 leads to a percent reduction of approximately 46% relative to the uninfected model, whereas an α of 1.0 leads to a percent reduction of approximately 50% (Fig. S4), so the model is not very sensitive to changes in α within biologically reasonable ranges. Altering α slightly modifies how quickly nest workers transition into the forager compartment depending on colony need (i.e. to make up the difference between the number of foragers present and the foraging cap κ). Smaller α indicates that the transition to fill the forager compartment occurs more slowly; this results in slightly fewer foragers being in the compartment leading to fewer individuals being at risk of becoming parasitized and thus a slightly reduced percent reduction in total colony size.

Supplementary Note 3: Additional Discussion Please refer to the main text for a complete discussion of the results of this study. Here we present additional discussion fodder for interested readers.

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The force of infection parameter β. The results of our model show that the force of infection parameter β is extremely important for predicting the severity of the impact of indirectly transmitting parasites on total colony size. The β parameter is composed of three different parameters: the rate at which individuals forage, the rate at which foragers encounter parasites in the extranidal environment while foraging, and the probability that an encounter with a parasite while foraging becomes a successful infection. For all three components of β, few if any empirical estimates are known. While work has been done on colony-wide foraging ranges and rates (77, 78), the frequency with which a given individual forages has been measured only a handful of times (79–82). Studies investigating individual foraging rates suggest that most foragers make few trips while a small proportion of foragers make many trips (79), which would lead to heterogeneity in infection risk. It is also important to acknowledge that non-foragers can also be periodically exposed to the extranidal environment and thus to parasite exposure. For example, in nomadic ant species such as those in the army ant tribe Ecitonini, frequent nest relocation could expose brood and intranidal workers as well. How such host ecological traits relate to parasitism pressure will be the subject of future work. The second component of β, how frequently parasites are encountered in the extranidal environment, remains enigmatic. For the vast majority of parasites infecting social insects, we have no idea how frequently potential hosts encounter infective stages such as fungal spores, largely due to the inherent difficulties in gathering such data. Some work has investigated the attack rates of parasitoids and found that attack rates can vary with host density and distance from nests (83) as well as with the number of host species in the area (84). For the ant fungal parasite Ophiocordyceps unilateralis s.l., infected cadavers have been behaviorally manipulated to die above foraging trails at the doorstep of the colony (54), where they can rain spores down on foragers as they leave the nest. However, these cadavers have been shown to have temporal windows over which spores are released and as foragers continue to walk over trails spores are effectively ‘cleaned up’, suggesting variable individual encounter rates. Thus, while information on actual encounter rates is scarce, heterogeneity appears to be likely in how frequently social insect hosts encounter parasites in the extranidal environment. The third component of β - how often parasite encounters lead to successful infection - is also relatively unknown, but host behavior and the effects of social immunity could serve to significantly reduce actual parasitism rates. When parasitoid flies are present, ants have been shown to reduce their foraging behaviour in response (58, 85, 86) and leaf-cutter ants even have workers who defend foragers from parasitoid attacks (87). For individuals that have picked up fungal spores or infective larval stage helminths, allogrooming and self-grooming can be effective strategies to prevent actual infection (24, 60, 88–90). Other mechanisms of social immunity, such as the spraying of formic acid and use of metapleural gland secretions (91, 92) can also kill parasites before infection can commence. Taken together, what information we do have on the above components of β suggests that the infective pressure of parasites is probably quite heterogeneous. Thus, our model, which uses a constant rate of parasite infection likely overestimates the impact of indirect parasite pressure on the reduction of colony population numbers. Better empirical estimates of the components of β are needed to truly identify the impact ex-nido parasites might have on colony populations.

Impacts of worker loss on colony functioning and fitness. This work has explored how the presence of indirect parasites could impact colony population dynamics, but how such potential population reductions translate into impacts on colony fitness remains an open question. Does a 10% reduction in colony population lead to a 10% reduction in colony fitness? This seems unlikely but empirical data are scant. Also, are colonies able to functionally buffer the loss of individuals through redundancies in their social organization? Indirect parasites, by modifying host behavior (i.e. reduced foraging in the presence of parasitoids (58), can also potentially impact colonies in nuanced ways that are subtler than simple reductions in colony size. Better understanding of the link between colony size, social structure, functioning, and fitness will allow us to better understand the interaction of social insects and their parasites in an evolutionary context. The impact of worker loss might manifest in other ways besides colony growth or the production of reproductives. Increasing evidence shows that there are important individual differences in task performance and personality (93), and these differences can cascade up to alter colony-level organization and phenotypes (94, 95). Thus, the parasite-induced loss of ‘keystone’ individuals could potentially alter colony phenotype, with downstream consequences for colony competitive ability and thus growth, survival, and reproduction. An additional way that the parasite-induced loss of workers could impact colonies is by promoting a positive feedback between worker loss and future parasitism events if, for example, inexperienced foragers had to make more trips or encountered more parasites in the environment than the experienced foragers that they replaced. While our model does not include this potential feedback, it would be interesting to examine both theoretically and empirically.

Impacts of parasites that cause morbidity rather than mortality. For modeling simplicity, we have only included parasite records where the parasite clearly caused mortality for the infected individual. We have excluded parasites that cause sickness behavior rather than mortality, or where the exact extent of host pathology is unclear. The colony-level impact of these parasites causing morbidity merits further investigation.

Need for more empirical measurements of ant colony sociometry. The modeling approach employed in this work has been useful for clarifying what host biological features might be the most important for colony survival while in the presence of

Quevillon et al. | Modeling ant parasites bioRχiv | 13 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. indirect parasites. While some parameter estimates were easily gleaned from published studies (i.e. brood developmental time), other important parameters such as colony birth rate and the proportion of the colony work force dedicated to foraging have far less empirical data available. In order to more accurately model how colonies grow in the presence of absence of parasitism and to validate this current model, we need better estimates of basic ant colony sociometry, as has been advocated by Tschinkel (96).

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Supplementary Note 4: Supplemental Figures and Tables

Fig. S1. Recovery dynamics from the uninfected model in the face of perturbation. Recovery dynamics (from numerical simulation) for the uninfected model (eqs. 1-3) under baseline parameter values given in Table 1. Initial colony conditions were set at Fs = 200 ants, N = 500 ants, and B = 100 ants and the colony was allowed to go to equilibrium. At t = 5,000 days, the colony size was reduced by 50%, 80%, or 90% (applied across all compartments) and allowed to return to equilibrium.

Recovery from colony perturbation in the uninfected model (β = 0 days-1)

5,000 perturbation

4,000

3,000

(ants) 2,000

Total colony size Total 1,000 Colony reduction 90% 80% 50% 0 0 2,000 4,000 6,000 8,000 10,000 Time (days)

Quevillon et al. | Modeling ant parasites bioRχiv | 15 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. S2. Sensitivity of model predictions to changing the natural mortality rate, µ. Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under changing values of µ from 0 to 0.1 days−1. (b) Total colony size (Fs + N + B) under changing values of µ from 0 to 0.1 days−1.

Sensitivity of model predictions to changing

100 the natural mortality rate, μ

75 μ = 1/730 days-1 1/365 days-1 1/182.5 days-1 1/91.25 days-1 1/30.42 days-1

50 relative to uninfected model

Percent colony size reduction 25

0 0 0.025 0.050 0.075 0.100

a Natural mortality rate, μ (days-1)

4,000

3,000 μ = 1/730 days-1 1/365 days-1 1/182.5 days-1 1/91.25 days-1 1/30.42 days-1

2,000 Total colony size

1,000

0 0 0.025 0.050 0.075 0.100 b Natural mortality rate, μ (days-1)

16 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. J Conditions for colony collapse

Fig. S3. Sensitivity of model predictions to changing the parasite-induced mortality rate, γ. Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under changing values of γ from 0 to 1 days−1. (b) Total colony size (Fs + N + B) under changing values of γ from 0 to 1 days−1.

Sensitivity of model predictions to changing parasite-induced mortality rate, γ 100 γ = 1/30 days-1 1/21 days-1 1/14 days-1 75 1/7 days-1

50

25 relative to uninfected model Percent colony size reduction

0 0 0.25 0.50 0.75 1.0 a Parasite-induced mortality rate, γ (days-1)

4,000

3,000 γ = 1/30 days-1 1/21 days-1 1/14 days-1 1/7 days-1 2,000 Total colony size Total 1,000

0 0 0.25 0.50 0.75 1.0 b Parasite-induced mortality rate, γ (days-1)

Quevillon et al. | Modeling ant parasites bioRχiv | 17 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. S4. Sensitivity of model predictions to changing the nest worker – forager transition term, α. Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under changing values of α from 0 to 1. (b) Total colony size (Fs + N + B) under changing values of α from 0 to 1.

Sensitivity of model predictions to changing the

100 nest worker - forager transition rate, α

75

baseline value 50

25 relative to uninfected model Percent reduction in colony Percent reduction size

0 0 0.25 0.50 0.75 1.0 a Nest worker - forager transition rate, α

4,000

3,000

baseline value

2,000

Total colony size Total 1,000

0 0 0.25 0.50 0.75 1.0

b Nest worker - forager transition rate, α

18 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. J Conditions for colony collapse

Fig. S5. Impact of seasonality in parasite force of infection on model predictions (one annual peak). Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under different maximum values of β with one annual peak. (b) Total colony size (Fs + N + B) under different maximum values of β with one annual peak.

Impact of seasonality in parasite force of infection: 1 annual peak

1.00 maximum β values 0.001 0.005 0.01 0.05 0.75 0.1 0.5 1

0.50 Force of infection, β 0.25

0 0 100 200 300 365 a Time (days)

4,000 no parasitism seasonal parasitism

maximum β values 0.001 3,000 0.005 0.01 0.05 0.1 0.5 1 2,000 Total colony size Total 1,000

0 0 5,000 10,000 15,000 Time (days) b

Quevillon et al. | Modeling ant parasites bioRχiv | 19 bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license.

Fig. S6. Impact of seasonality in parasite force of infection on model predictions (two annual peaks). Model predictions are from numerical simulation under baseline parameter values given in Table 1. (a) The proportion reduction in colony size relative to the uninfected model under different maximum values of β with two annual peaks. (b) Total colony size (Fs + N + B) under different maximum values of β with two annual peaks.

Impact of seasonality in parasite force of infection: 2 annual peaks

1.00

maximum β values 0.001 0.005 0.75 0.01 0.05 0.1 0.5 1 0.50 Force of infection, β 0.25

0 0 100 200 300 365 Time (days)

a

4,000 no parasitism seasonal parasitism

maximum β values 0.001 0.005 0.01 3,000 0.05 0.1 0.5 1

2,000 Total colony size Total 1,000

0 0 2,500 5,000 7,500 10,000 Time (days)

b

20 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. J Conditions for colony collapse

Colony birth rate, λ Ant species Birth rate Notes Reference Solenopsis invicta 0 - 50 workers/day Varies seasonally. (97) Lasius neglectus 6 - 14 eggs/day/queen - (98) Solenopsis invicta 62 - 1776 eggs/day Depends on the number of larvae already in the nest (47) Leptothorax acervorum 1 - 23 eggs/day - (99) Dolichoderus mariae 27 - 40 eggs/day/queen - (100) Dolichoderus mariae 1500 - 6000 eggs/day/nest - (100) Iridomyrmex purpureus 93 - 175 eggs/day/queen Depends on queen’s rank dominance (101)

Table S1. Parameter estimates from the literature: colony birth rate

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Brood developmental rate 1/φ Ant species Avg. time to adult emergence (days) Range (days) Reference Notes Solenopsis invicta 18 - (102)- Linepithema humile - 40 - 140 (103)- Solenopsis invicta - 23 - 55 (104)- Solenopsis invicta - 20 - 31 (105)- Dinoponera quadriceps 95 90 - 100 (106) within (107) Myrmecia regularis - 150 - 180 (108) within (107) Tetraponera anthracina 100 - (109) within (107) Aenictus laeviceps 65 - (110) within (107) Eciton burchelli 45 - (64) within (107) Eciton hamatum 50 - (110) within (107) Atta sexdens - 40 - 60 (111) within (107) Messor aciculatus 69.3 - (112) within (107) Messor pergandei 60 - (113) within (107) Myrmica rubra 115 - (114) within (107) Myrmica rubra 84 - (114) within (107) Myrmica rubra 54 - (114) within (107) Monomorium pharaonis 36.4 - (115) within (107) Monomorium pharaonis - 25 - 54 (116) within (107) Pogonomyrmex sp. - 35 - 42 (117) within (107) Solenopsis invicta - 24 - 25 (118) within (107) Solenopsis invicta - 55 (104) within (107) Tetramorium caespitum - 43 - 63 (119) within (107) Linepithema humile 44 - (120) within (107) Wasmannia auropunctata - 35 - 40 (121) within (107) Liometopum apiculatum 28.2 - (122) within (107) Liometopum apiculatum 70.8 - (122) within (107) Camponotus clariothorax 67 - (123) within (107) Camponotus festinatus 69 - (123) within (107) Camponotus laevigatus - 48 - 70 (123) within (107) Camponotus modoc 55 - (123) within (107) Camponotus planatus 57 54 - 58 (123) within (107) Camponotus sericeus - 24 - 26 (124) within (107) Camponotus sericeus 55 - (124) within (107) Camponotus sericeus 20 - (124) within (107) Camponotus vicinus - 54 - 70 (123) within (107) Formica polyctena - 35 - 45 (125) within (107) Formica rufa - 35 - 37 (126) within (107) Oecophylla longinoda 39 - (127) within (107) Prenolepis imparis - 70 - 90 (128) within (107)

Table S2. Parameter estimates from the literature: brood developmental rate

22 | bioRχiv Quevillon et al. | Modeling ant parasites bioRxiv preprint doi: https://doi.org/10.1101/470575; this version posted November 14, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. J Conditions for colony collapse

Proportion of the colony that forages, κ Ant species % that are foragers Notes Reference Formica polyctena 4.5 - 57.5% - (129) Solenopsis invicta 16 - 58% - (39) Pogonomyrmex owyheei 16% - (130) Pogonomyrmex badius 5 - 42% - (42) Solenopsis invicta 10 - 90% - (131) Solenopsis invicta 30 - 80% - (131) Pogonomyrmex mendozanus 10 - 13% - (132) Pogonomyrmex inermis 15 - 44% - (132) Pogonomyrmex rastratus 7 - 10% - (132)

Table S3. Parameter estimates from the literature: proportion of the colony foraging

Natural mortality rates, µ, of workers (not reproductives) Ant species Avg. life span (days) Notes Reference Atta colombica 3 - 4 days Avg. lifespan for 50% surviving (133) Cataglyphis bicolor 6 days - (134) Pogonomyrmex owyheei 14 days - (130) Pogonomyrmex sp. 30 days - (135) Solenopsis invicta 60 - 540 days - (136) Solenopsis invicta 70 - 490 - (137) Oecophylla smaragdina 175 - 210 days Avg. lifespan for 50% surviving (138) Aphaenogaster rudis > 1095 days Maximum recorded. (139), within (10). Leptothorax lichtensteini 912 days - (140), within (10). Leptothorax nylanderi 1095 days - (140), within (10). Monomorium pharaonis 63 - 70 days - (115), within (10). 620 days - (141), within (10). Myrmecia nigriceps 803 days - (141), within (10). Myrmecia nigrocinta 438 days - (141), within (10). Myrmecia pilosula 474 days - (141), within (10). 693 days - (141), within (10). Rhytidoponera purpurea 949 days - (141), within (10). Platythyrea punctata 100 - 200 days - (142) Lasius niger 325 days Avg. lifespan for 50% surviving, control treatment (143) Diacamma rugosum 208.8 days - (144)

Table S4. Parameter estimates from the literature: natural mortality rate

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Parasite-induced mortality rate, γ Parasite species Trans. loca- Avg. development Notes Reference tion time Ophiocordyceps unilateralis Ex-nido 6 - 20 days - (31) In-nido 3 - 7 days - (71) Metarhizium anisopliae In-nido 10 - 30 days Varies based on colony genetic di- (145) versity: polygynous vs. monogy- nous Metarhizium anisopliae In-nido 3 - 9 days - (89) Pseudacteon litoralis Ex-nido 46 days For larval + pupal development in- (44) side ant Pseudacteon litoralis Ex-nido 30 - 49 days For larval + pupal development in- (146) side ant Pseudacteon tricuspis Ex-nido 23 - 50 days For larval + pupal development in- (146) side ant Pseudacteon browni Ex-nido 23 - 43 days For larval + pupal development in- (146) side ant Pseudacteon crawfordi Ex-nido 24 - 32 days For larval + pupal development in- (146) side ant

Table S5. Parameter estimates from the literature: parasite-induced mortality rates

Foraging rate, a component of β Ant species Avg. foraging Notes Reference trips/day per ant Pogonomyrmex barbatus 1 - 25 (24 hrs) - (82) Paraponera clavata 1 - 9 (12 hrs) - (80) Cataglyphis bicolor 3.7 (24 hrs) - (81) Pogonomyrmex montanus 50 - 950 trips/day per Reported as mean foraging trips per (117) colony day for entire colonies, not for indi- vidual foragers. Number of forag- ing trips/day/colony varies season- ally. Pogonomyrmex subnitidus 50 - 10,500 trips/day Reported as mean foraging trips per (117) per colony day for entire colonies, not for indi- vidual foragers. Number of forag- ing trips/day/colony varies season- ally. Pogonomyrmex rugosus 50 - 25,000 trips/day Reported as mean foraging trips per (117) per colony day for entire colonies, not for indi- vidual foragers. Number of forag- ing trips/day/colony varies season- ally.

Table S6. Parameter estimates from the literature: foraging rates

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Parasite encounter rate, a component of β Ant species Parasite Attack rate Reference Linepithema sp. Pseudacteon sp. No direct attack rates provided, but (84) see results section. Azteca instabilis Pseudacteon sp. 0 - 8 attacks/15 mins (147) Atta vollenweideri, Acromyrmex sp. Various 0.3 - 1.2 attacks/minute (83)

Table S7. Parameter estimates from the literature: parasite encounter rates

Quevillon et al. | Modeling ant parasites bioRχiv | 25